The modified, discrete, Lévy-transformation is Bernoulli. (English) Zbl 0761.60043

Dubins, Lester E.; Smorodinsky, Meir

Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 157-161 (1992).

Summary: [For the entire collection see Zbl 0754.00008.]
From the absolute value of a martingale, \(X\), there is a unique increasing process that can be subtracted so as to obtain a martingale, \(Y\). Paul Lévy discovered that if \(X\) is Brownian motion, \(B\), then \(Y\), too, is a Brownian motion. Equivalently, Lévy found that the transformation that maps \(B\) to \(Y\) is measure-preserving. Whether it is ergodic, a question raised by Marc Yor, is open. Here, the natural analogue of Lévy’s transformation for the symmetric random walk is modified and, thus modified, is shown to be measure-preserving. The ergodicity of this transformation is then established by showing that it is isomorphic to the one-sided, Bernoulli shift-transformation associated with a sequence of independent random variables, each uniformly distributed on the unit interval.

MSC:

60G48

Generalizations of martingales

Keywords:

martingale; Brownian motion; Lévy’s transformation for the symmetric random walk; ergodicity; Bernoulli shift-transformation

Citations:

Zbl 0754.00008